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On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself

  Published:2006-06-01
 Printed: Jun 2006
  • Jan J. Dijkstra
  • Jan van Mill
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Abstract

In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\R$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\Q$ onto itself is homeomorphic to the infinite power of $\Q$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\Q$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to $\R^n$ for $n\ge 2$, by linking the groups in question with Erd\H os space.
Keywords: homeomorphism group, real line, countable dense set, pseudoboundary, Erd\H{o}s space, hyperspace homeomorphism group, real line, countable dense set, pseudoboundary, Erd\H{o}s space, hyperspace
MSC Classifications: 57S05 show english descriptions Topological properties of groups of homeomorphisms or diffeomorphisms 57S05 - Topological properties of groups of homeomorphisms or diffeomorphisms
 

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